The probability of a temple facing East
So this is what I’ve been working on this week. I’ve been looking at the orientations of Greek temples. There is an idea that Greek temples always face east, and that’s what I’m testing at the moment. If I can show that Greek temples do face East then things get interesting. This is because in Sicily in the first millennium BC the natives take on a lot of Greek material. If I can show that the natives are still practising their own religions in their own way, then I have strong argument that they’re using Greek pottery and so on for their own purposes rather than simply becoming Greeks themselves. I have results and I’m trying to put them together meaningfully.
A lot of the significance depends on the data set and how I use it. For instance if I have only one temple and it points East, that doesn’t really mean a lot. It has to point somewhere, so why should that be special? It could face that way by chance. If I have two temples facing East then that’s a bit better, but it’s still hardly impressive. At it happens I have measurements for forty-two temples, but not all of them face East. So are my results significant? Below is me trying to work this out and come up with a better answer than: “Yes, because I say so.” It follows quite a few other chapters in the thesis, so it might not all make sense, but it should make enough sense for people to point out any obvious mistakes in my handling of probability.
Some of the formulae may seem a little odd, but hopefully they’re clear enough. I’ll have to get to grips with MathML to generate some formulae graphics for the actual print. This is very much first draft material.
Given the orientation of any individual temple, it is impossible to say precisely purely from statistics what the motive for its orientation was. By its very nature any temple must have an opening facing in one direction and so who is to say that it was aligned intentionally rather than by chance? However, it is possible to quantify the effects of chance when the data are aggregated. For the first example I shall tackle the question “Do Greek Temples Face East?” For the sake of this question I am assuming any temple which faces between 0° through 90° to 180° is facing East and that any temple facing between 180° through 270° to 0° is facing west. I should also clarify that 0° is due North as Penrose took his 0° reference to be due South.
In the absence of any a priori assumptions, the chance of any individual temple facing East should be 0.5, where 0 is complete impossibility and 1 is absolute certainty. This is a well known problem in probability akin to coin tossing and so it is easy to calculate what, on average will be the result if forty-two temples are aligned purely by chance. The relevant formula is xbar = 0.5n. In this case n is forty-two and so on average twenty-one temples will face East. In these results forty-one temples face the eastern horizon, and so this would appear to be significant. The question usually asked is how significant is this?
The combinations of possible result expand rapidly as more temples are added to the test set. For example if we have just one temple in our test set then there are just two possible outcomes.
However if we add a second temple four outcomes are possible.
If we add a further temple to make a set of three then there are eight potential outcomes
To complete the example a fourth temple will provide sixteen potential combinations:
So the number of combinations of potential East-West orientations in a set of n temples is 2n. For the complete set of forty-two temples are 242 potential combinations by chance. This is a large figure: four trillion three hundred ninety-eight billion forty-six million five hundred eleven thousand one hundred and four combinations are possible. That all bar one of these will face East as in the results of the survey will only occur in forty-two of these combination as the aberrant temple could be the first, second, third…forty-second temple measured. Simplistically I could argue that the odds that these results were due to chance were 42/4,398,046,511,104 which reduces to 21/2,199,023,255,552. This is approximately one in a hundred million. This is often the way such odds are calculated in archaeology (note to self add several references). However there is a superficial flaw.
If all the temples had faced East then this also would have been considered significant. Therefore I should not be calculating the odds that forty-one temples faced East, but rather that at least forty-one temples faced East. This adds one further combination, namely EEEEEE…EEEEEE to the set. Perhaps it would be better to quote odds of 43 to 4,398,046,511,104 which is still approximately one in a hundred million. Yet this question can also be asked if fewer temples matched the tested alignment. What would this chapter have looked like if only forty temples had faced East?
There are considerably more combinations of where two temples in the set face West. The aberrant temples could be the first and second temples measured, or the first and third or the first and fourth, through to the forty-first and forty-second. There are in fact eight hundred and sixty-one combinations. This means that forty temples would be significant then the odds of similar results being due to chance are 904 to 4,398,046,511,104. This would still be an impressively improbable figure, but this line of reasoning raises the question of whether thirty-nine temples would be significant or thirty-eight. This is a considerably more serious flaw. Complicating the matter further had forty-one of forty-two temples pointed West it would be equally unlikely. Rather than decide what is significant a posteriori it would be more helpful if it were possible to decide a priori what would be considered significant.
The usual figure adopted in psychological tests, where results are prone to be products of chance, is to assume that anything below a 5% probability is significant. This is particularly suitable for application here as 5% is an easily calculable figure. Ninety-five per cent of results in a standard bell curve will be within two standard deviations of the mean. Calculation of the standard deviation of a binomial probability is easily calculated via a formula.
where n is the number of temples in the set, p is the probability of the temple facing the target range and q is the probability of the temple falling outside this range. This would be equal to 1-p.
For this example, if I were to build thousand of sets of forty-two temples, aligning them randomly, the average number of temples in each set facing East would be 21 and 95% of them would have 21+/-6.48074 temples. Therefore for the purposes of this study any result between fourteen and twenty-eight temples would have been considered significant.
This is unquestionably considerably less emotionally satisfying than announcing the probability of the results being due to chance are a hundred million to one but, given the small data set, it is considerably more defensible. The problem is that it is not sufficient that the results satisfy this one in twenty rule. Richard Feynman has argued that this rule would logically mean that one in twenty psychological laws were statistical flukes. This is not entirely fair, such results would subject to further tests in Psychology. However I simply apply this criterion to a series of tests then, if I have more than twenty tests, by sheer chance I should expect one ‘significant’ result. There is in particular the tendency to pursue tests until something significant is found. Feynman’s criticism would then apply to this work. Therefore I propose that this 95% criterion is only used as a filter. No matter how seductive a proposal is, if it does not meet this criterion I should conclude that the data is insufficient to support the hypothesis. If the proposal surpasses this criterion then there is still an imperative to provide a historical justification for the result. In this case the result, only one temple faces away from East appears compelling, but that means there is a need to explain why that temple – the Hekataion at Selinunte – is facing contrary to the other temples.
The Hekataion at Selinunte is dedicated to Hekate Triformis. Triformis refers to the three aspects of the goddess. She was Artemis, Selene and Hekate. In the form of Hekate she was the dark moon around the crescent. She was known as the opener of the door to Hades (check Cashford) and so played a role in the quest to recover Persephone from Hades. This is significant as the Hekataion is part of a complex at Selinunte, the other two major gods there being Demeter Malaphoros and Zeus Melikhios who both played a role in the mysteries of Eleusis. By facing the setting side of the sky, this orientation could be intentionally contrary to the typical cult practices. However it should also be noted that despite being a lunar goddess the temple points too far to the north to face the most northerly moonsets. This would therefore not be any form of celestial alignment but a conscious rejection of celestial alignment. This explanation is plausible without invoking new properties for Hekate and would be consistent with the local geography and context of the site.
It would seem justifiable to therefore claim that Greek temples typical face east.
The next part will follow tomorrow.